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In a previous paper, we constructed an explicit dynamical correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps on the Riemann sphere. In this paper, we show that their deformation spaces share many striking similarities. We establish an analogue of Thurston's compactness theorem for critically fixed anti-rational maps. We also characterize how deformation spaces interact with each other and study the monodromy representations of the union of all deformation spaces.
We prove uniform a priori bounds for Siegel disks of bounded type that give a uniform control of oscillations of their boundaries in all scales. As a consequence, we construct the Mother Hedgehog for any quadratic polynomial with a neutral periodic point.
Let ℂv be a characteristic zero algebraically closed field which is complete with respect to a non-Archimedean absolute value. We provide a necessary and sufficient condition for two tame polynomials in ℂv[z] of degree d≥2 to be analytically conjugate on their basin of infinity. In the space of monic centered polynomials, tame polynomials with all their critical points in the basin of infinity form the tame shift locus. We show that a tame map f∈ℂv[z] is in the closure of the tame shift locus if and only if the Fatou set of f coincides with the basin of infinity.
Let K be an algebraically closed and complete nonarchimedean field with characteristic 0 and let f∈K[z] be a polynomial of degree d≥2. We study the Lyapunov exponent L(f,μ) of f with respect to an f-invariant and ergodic Radon probability measure μ on the Berkovich Julia set of f and the lower Lyapunov exponent L−f(f(c)) of f at a critical value f(c). Under an integrability assumption, we show L(f,μ) has a lower bound only depending on d and K. In particular, if f is tame and has no wandering nonclassical Julia points, then L(f,μ) is nonnegative; moreover, if in addition f possesses a unique Julia critical point c0, we show L−f(f(c0)) is also nonnegative.
This paper studies polynomials with core entropy zero. We give several characterizations of polynomials with core entropy zero. In particular, we show that a degree d post-critically finite polynomial f has core entropy zero if and only if f is in the degree d main molecule. The characterizations define several quantities which measure the complexities of polynomials with core entropy zero. We show that these measures are all comparable.
Recent works [MMO1, arXiv:1802.03853, arXiv:1802.04423, arXiv:2101.08956] have shed light on the topological behavior of geodesic planes in the convex core of a geometrically finite hyperbolic 3-manifolds M of infinite volume. In this paper, we focus on the remaining case of geodesic planes outside the convex core of M, giving a complete classification of their closures in M. In particular, we show that the behavior is different depending on whether exotic roofs exist or not. Here an exotic roof is a geodesic plane contained in an end E of M, which limits on the convex core boundary ∂E, but cannot be separated from the core by a support plane of ∂E. A necessary condition for the existence of exotic roofs is the existence of exotic rays for the bending lamination. Here an exotic ray is a geodesic ray that has finite intersection number with a measured lamination L but is not asymptotic to any leaf nor eventually disjoint from L. We establish that exotic rays exist if and only if L is not a multicurve. The proof is constructive, and the ideas involved are important in the construction of exotic roofs. We also show that the existence of geodesic rays satisfying a stronger condition than being exotic, phrased in terms of only the hyperbolic surface ∂E and the bending lamination, is sufficient for the existence of exotic roofs. As a result, we show that geometrically finite ends with exotic roofs exist in every genus. Moreover, in genus 1, when the end is homotopic to a punctured torus, a generic one (in the sense of Baire category) contains uncountably many exotic roofs.
Submitted 15 August, 2024; v1 submitted 8 October, 2022; originally announced October 2022.
For a post-critically finite hyperbolic rational map f, we show that the Julia set Jf has Ahlfors-regular conformal dimension one if and only if f is a crochet map, i.e., there is an f-invariant graph G containing the post-critical set such that f|G has topological entropy zero. We use finite subdivision rules to obtain graph virtual endomorphisms, which are 1-dimensional simplifications of post-critically finite rational maps, and approximate the asymptotic conformal energies of the graph virtual endomorphisms to estimate the Ahlfors-regular conformal dimensions. In particular, we develop an idea of reducing finite subdivision rules and prove the monotonicity of asymptotic conformal energies under the decomposition of rational maps.
Submitted 27 September, 2022; originally announced September 2022.
In this paper we initiate the study of birational Kleinian groups, i.e.\ groups of birational transformations of complex projective varieties acting in a free, properly discontinuous and cocompact way on an open set of the variety with respect to the usual topology. We obtain a classification in dimension two.
| arXiv:2103.09350 |
In this paper, we develop a theory on the degenerations of Blaschke products d to study the boundaries of hyperbolic components. We give a combinatorial classification of geometrically finite polynomials on the boundary of the main hyperbolic componentd containing zd. We show the closure d⎯⎯⎯⎯⎯⎯⎯⎯ is not a topological manifold with boundary for d≥4 by constructing self-bumps on its boundary.
In this paper, we study quasi post-critically finite degenerations for rational maps. We construct limits for such degenerations as geometrically finite rational maps on a finite tree of Riemann spheres. We prove the boundedness for such degenerations of hyperbolic rational maps with Sierpinski carpet Julia set and give criteria for the convergence for quasi-Blaschke products d, making progress towards the analogues of Thurston's compactness theorem for acylindrical 3-manifold and the double limit theorem for quasi-Fuchsian groups in complex dynamics. In the appendix, we apply such convergence results to show the existence of certain polynomial matings.